Quantum Frobenius and its kernel in the good odd-order regime

By [F1], every divided power in $B$ vanishes unless its index is divisible by $\mathrm{\ell }$, and the remaining ones are ordinary divided powers of ${\bar{e}}_i,{\bar{f}}_i$. Every toral generator is a polynomial in the ${\bar{H}}_i$, with its exact formula given in [F1]; further relations in $B$ are not yet excluded. Thus ${\bar{e}}_i,{\bar{f}}_i,{\bar{H}}_i$ generate $B$. In the complete mixed formula at exponents $\mathrm{\ell },\mathrm{\ell }$, all intermediate contraction indices $0<t<\mathrm{\ell }$ have remaining divided factors of nonzero residue and vanish in $B$. Its two surviving terms give $[{\bar{e}}_i,{\bar{f}}_i]={\bar{H}}_i$; opposite colors commute. Iterating the toral shift through an $\mathrm{\ell }$th divided power and using $T_{i,-\mathrm{\ell }{a}_{ij},\mathrm{\ell }}=H_i-a_{ij}$ gives $[{\bar{H}}_i,{\bar{e}}_j]=a_{ij}{\bar{e}}_j$ and $[{\bar{H}}_i,{\bar{f}}_j]=-a_{ij}{\bar{f}}_j$. The ${\bar{H}}_i$ commute.

For $i\mathrm{\ne }j$, put $a=-a_{ij}$. Apply [F2] with $n=\mathrm{\ell }$, $m=(a+1)\mathrm{\ell }$. In $B$ only $r,s$ divisible by $\mathrm{\ell }$ survive. For such $r$, ${\zeta }_i^{r(a\mathrm{\ell }-(a+1)\mathrm{\ell }+1)}=1$ and $(-1{)}^r=(-1{)}^{r\mathrm{/}\mathrm{\ell }}$ because $\mathrm{\ell }$ is odd. By [F1] the resulting relation is ${\sum }_{b+c=a+1}(-1{)}^b{\bar{e}}_i^b{\bar{e}}_j{\bar{e}}_i^c\mathrm{/}(b!c!)=0$, precisely the ordinary Serre relation after multiplying by $(a+1)!$. The negative relation is identical with $\bar{f}$ in place of $\bar{e}$. Thus [F4]'s Serre presentation and step 1.1 give a surjective homomorphism $\psi :U(\mathfrak{g})\to B$. No presentation theorem for the divided-power specialization has been assumed.

Fix any dominant $\mu \in P$ and specialize the actual lattice $V_A(\mathrm{\ell }\mu )$ of [F3] at $q=\zeta$. Finite freeness gives a finite-dimensional module with a nonzero rank-one top weight and its actual integral weight decomposition; the highest vector still generates it, since $V_A=U_A^{-}v$. The toral character of a weight $\nu$ has $K_i$ value ${\zeta }_i^{\nu (h_i)}$ and $H_i$ value $\lfloor \nu (h_i)\mathrm{/}\mathrm{\ell }\rfloor$, by [F1] including negative upper indices. These values determine each integer $\nu (h_i)$: the first gives its residue and the second its quotient. Since simple coroot coordinates distinguish weights in a semisimple lattice, the full toral characters of distinct weights are distinct. Finite polynomial interpolation in commuting toral operators therefore projects onto each actual weight space and preserves every submodule. A simple quotient $M$ of this finite-dimensional cyclic module exists by choosing a proper submodule of maximal dimension. It retains a nonzero top vector $v$ of weight $\mathrm{\ell }\mu$, since a submodule containing $v$ would be the entire module; it also retains the weight decomposition and toral characters.

Every positive divided power kills $v$. For every $i$, also $F_{i}v=0$. Indeed for $j\mathrm{\ne }i$ the positive $j$th divided powers commute past $F_i$ and kill it on $v$. For color $i$, the complete mixed reordering gives $E_i^{(r)}{F}_{i}v=0$ for $r>1$, since the possible contraction has right factor $E_i^{(r-1)}v=0$; at $r=1$ its coefficient is $[\mathrm{\ell }\mu (h_i){]}_{{\zeta }_i}=0$. Hence, if $F_{i}v$ were nonzero, it would be a highest-weight vector of weight $\mathrm{\ell }\mu -{\alpha }_i$. Triangular spanning then puts its entire generated submodule in weights at most this weight, so it misses the original top weight and is proper. Simplicity of $M$ excludes this. Thus all small generators $E_i,F_i$ kill $v$.

Let $N\subseteq M$ be the common kernel of all $E_i,F_i$. It contains $v$. It is a sum of actual weight spaces: each small operator shifts the actual weight by its simple root, and distinct input weights give distinct output weights. Since the torus acts diagonally on those spaces, $N$ is toral-stable. It is also stable under every $E_i^{(\mathrm{\ell })},F_i^{(\mathrm{\ell })}$. For same-sign distinct colors, the good-order assumptions give $\mathrm{\ell }>-a_{ij}$: finite-type off-diagonal entries have absolute values at most $3$, odd order is at least $3$, and the only possible equality $3$ in $G_2$ is excluded by $3\nmid \mathrm{\ell }$. Apply [F2] with $n=1,m=\mathrm{\ell }$ to a vector of $N$. Its $r=0$ term is $E_j{E}_i^{(\mathrm{\ell })}$ applied to that vector; the $r=\mathrm{\ell }$ term has rightmost $E_j$, and every intermediate term has rightmost $E_i^{(s)}$ with $0<s<\mathrm{\ell }$. These all vanish by [F1], leaving the desired zero; the negative proof is identical. Same-color same-sign operators commute. Opposite distinct colors commute. For opposite signs in the same color, mixed reordering gives an error factor of degree $\mathrm{\ell }-1$, which vanishes on $N$ after its toral factor acts diagonally; hence these small operators also annihilate the image of $N$ under the big divided generator. Finally [F1] generates all divided powers from the small and big generators. Thus $N$ is a full submodule, and simplicity yields $N=M$.

The cyclic module $M$ is now generated from $v$ by big divided generators and the torus alone, so all its weights differ from $\mathrm{\ell }\mu$ by multiples of $\mathrm{\ell }$ times roots. Every $K_h$ therefore acts as $1$, and step 5.1 shows that $J$ kills $M$. Along $\psi$, it is a finite-dimensional simple classical module, since $\psi$ is onto, with highest vector $v$ and ${\bar{H}}_{i}v=\mu (h_i)v$ by [F1]. Classification in [F4] identifies it with $L(\mu )$. Consequently every classical dominant simple with highest weight in $P$ occurs as a $\psi$-pullback. This does not assert simplicity of the specialized Weyl module before taking its quotient.

These classical modules separate $U(\mathfrak{g})$. For completeness, set $V=\mathfrak{g}$ and let $\rho :\mathfrak{g}\to \mathrm{End}(V)$ be its faithful adjoint representation; semisimplicity makes its kernel the zero center. On the polynomial algebra of the matrix space $\mathrm{End}(V)$, define the linear vector field $D_x$ by $D_x(M)=M\rho (x)$. On coordinate functions its bracket is $[D_x,D_y](M)=M[\rho (x),\rho (y)]$, so this is a Lie representation and extends to $U(\mathfrak{g})$. If $u\mathrm{\ne }0$ has PBW degree $m$, its top symbol is nonzero in ${\mathrm{Sym}}^m\mathfrak{g}$ by [F4]. At the identity matrix the principal symbol of $D_u$ is its image under ${\mathrm{Sym}}^m\rho$, which is injective because $\rho$ is injective and a finite basis of its image extends to a basis of the matrix space. Choose a homogeneous degree-$m$ polynomial in the shifted matrix coordinates, paired nontrivially with this symbol. Its lower derivatives vanish at the identity, so $D_{u}p$ has nonzero value there. Thus $u$ acts nontrivially on this polynomial representation.

Polynomials of bounded total degree form finite-dimensional stable subspaces for these linear vector fields. By [F4] each decomposes into classical simples. All its weights lie in the root lattice: the adjoint weights, their duals and their tensor or symmetric powers have this property. Hence every highest weight of these constituents belongs to $Q_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}\subseteq P$. If an element killed every simple supplied by step 6.1, it would kill every finite polynomial subspace and hence the entire polynomial representation, contradicting step 7.1. The intersection of their annihilators is therefore zero. Since $\ker \psi$ annihilates all of them, $\psi$ is injective and hence an isomorphism. If $\mathfrak{g}=0$, $U(\mathfrak{g})=\mathbb{C}$ acts faithfully on the constant polynomial module, giving the same conclusion directly.

Compose the quotient map $U_{\zeta }^L\to B$ with ${\psi }^{-1}$. This constructs a surjective algebra map with exact kernel $J$. Its divided-power values follow from [F1], and its toral values follow from the toral quotient formula there. These formulas cover every generator of the actual integral form, so the map is unique. In particular this construction proves well-definedness on all relations of the actual specialized subalgebra, rather than checking only a possibly incomplete proposed list of divided-power relations.

The actual root Hopf structure in [F1] makes $J$ a Hopf ideal. On its generators, $\mathrm{\Delta }(E_i)$ and $\mathrm{\Delta }(F_i)$ lie in $J\otimes U+U\otimes J$; $\mathrm{\Delta }(K_h-1)=(K_h-1)\otimes K_h+1\otimes (K_h-1)$; and counit and antipode preserve their ideal. In $B$, the interior terms of the divided-power coproduct of $E_i^{(\mathrm{\ell })},F_i^{(\mathrm{\ell })}$ vanish by [F1], and $K_h=1$, so ${\bar{e}}_i,{\bar{f}}_i$ are primitive. Their commutators ${\bar{H}}_i$ are primitive as well. Therefore $\psi$ preserves the usual classical Hopf maps on all its generating Lie elements and is a Hopf isomorphism. Its inverse composed with the quotient is the claimed Hopf Frobenius map. The inherited Choice assumptions are exactly [C1]; no rank-one restriction, integral-presentation shortcut, or unidentified small-quantum-group kernel was used.